Apollo was built in the era of "big iron" computer mainframes. The concept of an onboard computer for space missions was radically new. There was only one place in the world that could design the Apollo Guidance Computers, and that was MIT. It took most of the U.S.'s production of integrated circuits, another radical idea replacing individual vacuum tubes, to build the AGC's. It took 2,000 person-years of independent code review and validation to ensure that they would operate properly.
Besides the technical challenges, the Apollo missions were a matter of National Prestige. In the Space Race between the United States and the Soviet Union, President Kennedy said we were going to the Moon and return safely before the end of the (1960's) decade, so we did.
The flight computers for the Saturn launch vehicle were an evolution of earlier missile guidance efforts. The massive Saturn-V first stage was built from clustered Jupiter rockets. The earlier and smaller Saturn-I was built of clustered Redstone rockets, which were themselves a derivative of the early German V-2 rockets by the Von Braun Team. The Saturn's upper stages were all new technology, using liquid hydrogen and oxygen. The vehicle had to achieve and maintain a precise trajectory from its launch site in Florida, to lunar orbit, to the lunar surface and back, and then return to Earth. All this took unprecedented computing power.
This book is a brief synopsis of the architecture and applications of the Apollo computers. There is a lot of archived material on the topic, and the list of references in this book is a good starting point. The amazing thing is, the first computer on another world was designed and built before computers were commodity items. It worked as planned. Although your phone has more computing power than the entire Saturn vehicle, you shouldn't trust it to get you to the moon and back.
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